Approximation

An approximation is anything that is intentionally similar to something else without being exactly equal. The word comes from the Latin term approximatus, which stems from proximus, meaning "very near." In technical or scientific fields, experts use terms like approximate or approximation to describe values or properties that are nearly correct. In everyday English, we often use simpler words like "roughly" or "around" to convey the same idea. For example, saying the time is "around 10 o'clock" is an approximation. This concept can apply to many different things, including numbers, shapes, mathematical functions, and physical laws.

In mathematics, approximation theory is a formal branch of study. It is considered a quantitative part of functional analysis. One specific area is Diophantine approximation, which focuses on approximating real numbers using rational numbers. Mathematicians use approximations when an exact numerical value is unknown or too difficult to obtain. Sometimes, a known form can represent a real form so well that no significant deviation is found. This allows researchers to work with complex systems by using models that are close enough to the truth.

Numerical approximations often involve the use of significant digits. For instance, the notation 1.5 × 10^6 indicates a value near 1,500,000. This specific notation implies the true value is between 1,450,000 and 1,550,000. In contrast, the notation 1.500 × 10^6 is much more precise. It implies the true value is between 1,499,500 and 1,500,500. Using a small number of digits can lead to rounding errors or other calculation errors. Most tools, such as calculators, slide rules, and log tables, produce approximate answers for complex calculations. Even computer results are usually approximations expressed in a limited number of significant digits.

Mathematics also uses symbols to denote these relationships, though notation is not always consistent. The symbol ≈ is commonly used to mean "approximately equal to." This specific symbol was introduced in a slightly different version by the British mathematician Alfred Greenhill in 1892. He included it in his book, Applications of Elliptic Functions. Other symbols include ~ for asymptotic equality or function proportionality. In LaTeX, different symbols carry specific meanings, such as ≃ for function asymptotic equivalence or ≅ for figure congruence. Because different texts use these symbols differently, the meaning can vary depending on the source.

In science, approximation is a natural part of experiments and theories. A scientific theory might predict something that differs from an actual measurement. This happens because real-world factors, like air resistance, might not be included in a simple model. Under these conditions, the theory serves as an approximation of reality. Furthermore, limitations in measuring techniques can cause measurements to be approximations of actual values. The correspondence principle suggests that new scientific theories should reproduce the results of older theories in areas where those old theories worked well. In this view, the old theory becomes an approximation of the new, more accurate theory.

Physicists often use approximations to manage extreme complexity. For example, they might approximate the shape of the Earth as a sphere. While this is not perfectly accurate, it makes calculating characteristics like gravity much easier. Another complex example is the motion of planets orbiting a star. To solve this, scientists can use a process called iterations. In the first iteration, they might ignore gravitational interactions between planets and assume the star is fixed. In the next iteration, they add the first-order gravity interaction from each planet to the others. This process repeats until the solution is sufficiently precise. These methods, including the use of perturbations or simulations, help manage the "three body problem."

Beyond math and science, approximation has a specific meaning in law within the European Union. In this context, it refers to the process of incorporating EU legislation into the national laws of Member States. This happens despite the different legal frameworks in each country. Approximation is a requirement for new states joining the EU and is a continuing process for existing members. For example, the Trade Marks Directive of 2015 was designed to approximate the laws of Member States regarding trade marks. The European Commission describes this legal approximation as a unique obligation of membership in the Union.